On the relation of three theorems of functional analysis to the axiom of choice

In the following, we prove by extending methods of Rhineghost, that both the Arzel\`a-Ascoli theorem and the Fr\'echet-Kolmogorov theorem are equivalent to the axiom of countable choice for real sets. We further present supposedly new proofs for the facts that the uniform boundedness principle implies the axiom of countable multiple choice, and that the axiom of countable choice implies the uniform boundedness principle, show equivalence of the axiom of countable multiple choice to a weak version of the uniform boundedness principle, and prove that the uniform boundedness principle implies the axiom of countable partial choice for sequences of sets with cardinality bounded by a natural number. Along the way, we also give a proof for a Tietze extension theorem-like result for equicontinuous function sets. That proof may be new.